Tulio Eduardo Restrepo Medina RADIATIVELY INDUCED VECTOR REPULSION FOR LIGHT
Progress On Studying Uncertainties Of Parton Distributions ... · Heavy Quark Distributions: Very...
Transcript of Progress On Studying Uncertainties Of Parton Distributions ... · Heavy Quark Distributions: Very...
Progress On Studying Uncertainties Of Parton Distributions And Their Physical Predictions
Brief summary of traditional global QCD analysis What’s uncertain about parton distribution functions?
Studying uncertainties of parton distributions and their physical predictions: Ideal approaches in a world with perfect experiments Progress on a realistic global analysis of uncertainties • General approach and techniques � two complementary methods: (J. Pumplin, D. Stump)
• The Hessian method: orthonormal basis PDF sets, and a general master equation for calculating uncertainties of any quantity dependent on PDFs
• The Lagrange multiplier method: robust and optimized PDFs for quantifying uncertainties of specific physical variables
Results and physical applications • W/Z cross-sections at Tevatron/LHC/RHIC • W rapidity distribution, (W-mass measurement) • Parton Luminosities at Tevatron/LHC/RHIC/VLHC
(Higgs-, top-X-sections, high pT jets, ... etc.) Outlook
Global QCD Analysis in a Nutshell
Master Equation for QCD Parton Model
{ the Factorization Theorem
F�A(x;m
Q;M
Q) =X
a
faA(x;m
�) F̂�a (x;
Q
�;M
Q)+O((
�
Q)2)
FA
A
a
fAa
A
Fa^
H H
Sources of Uncertainties and Challenges:
� Experimental errors;
� Parametrization dependence;
� Higher-order corrections; Large Logarithms;
� Power-law (higher twist) corrections.
ExperimentalInput
Parton Dist. Fn.Non-Perturbative Parametrization at Q0
GLAP Evolution to Q
Hard Cross-sectionperturbative calculable
(may contain snLogn(M/Q))
(and uncertainties on errors!)
*, W, Z
q
g q
q
q
*, W, Z
g
q q
q
*, W, Z
(dir)
DIS
DY
Dir.Ph.
Jet Inc.
e N N
N N
p N N
k Np N
p N N
k Np N
p p
SLACBCDMS
NMC, E665H1, ZEUS
CDHS, CHARMCCFR
CHORUS
E605, E772NA51E866
CDF, D0
WA70, UA6E706
CDF, D0
CDF, D0
Experimental Input: Phyical Processes & Experiments
100
101
102
103
104
1/X
100
101
102
Q (GeV)
DIS
(fix
ed ta
rget
)H
ER
A (
’94)
DY
W-a
sym
met
ryD
irect
-γJe
ts
Th
e k
ine
ma
tic r
an
ge in
th
e (
x, Q
) p
lan
e o
f d
ata
po
ints
incl
ud
ed
in
a typ
ica
l glo
ba
l QC
D a
na
lysi
s –
CT
EQ
5
Global QCD Fit (CTEQ)
* Parametrization of the non-perturbative PDFs:(at Q0 = 1 GeV), e.g.
fi(x, Q0) = ai0x
ai1(1− x)a
i2(1 + ai
3xai4).
* The fitting is done by minimizing a global effec-tive “chi-square” function, χ2global,· overall figure of merit of the fit· regulates trade-off’s between the many expts.
χ2g =∑n
χ2n (n labels the experiments)
χ2n =
(1− Nn
σNn
)2+ wn
∑I
(NnDnI − TnI(a)
σDnI
)2
DnI : data point σDnI: combined error
TnI(a): theory value (dependent on {ai})for the Ith data point in experiment n.
Nn: Normalization para. for expt. n.wn: a “prior” based on physics considerations
(1: inclusion; 0: exclusion; or other).
* This effective χ2 function does not have thefull probabilistic significance of an ideal statisticalanalysis. (cf. uncertainty section.)
x
0
0.2
0.4
0.6
0.81
1.2
x f(x,Q)
10-4
10-3
10-2
10-1
.2.3
.4.5
.6.7
Q =
5 G
eV
.8
Ove
rvie
w o
f Par
ton
Dis
trib
utio
n F
unct
ions
of t
he P
roto
n
CT
EQ
5MG
luon
/ 15
d bar
u bar
s c u v d v (dba
r-u b
ar)
* 5
CTEQ5M �t to E866 :�pd
2�pp(x2)
0.0 0.1 0.2 0.3X2
0.7
0.8
0.9
1.0
1.1
1.2
1.3
Xse
c(pd
)/2X
sec(
pp)
E866CTEQ5M
Fit to CDF W-lepton asymmetry:
0.5 1 1.5 2
Y
-0.1
0
0.1
CD
F W
-lept
on A
sym
met
ry
CTEQ5MCDF data
5010
015
020
025
030
035
040
0p T
(G
eV)
0
0.2
0.4
0.6
0.81
1.2
1.4
Incl
. Jet
: p
t7 * d
σ/dp
t
(Err
or b
ars:
sta
tistic
al o
nly)
14%
< C
orr.
Sys
. Err
. < 2
7%
Rat
io: P
rel.
data
/ N
LO Q
CD
(C
TE
Q5M
| C
TE
Q5H
J)
norm
. fac
or :
CT
EQ
5HJ:
1.04
CT
EQ
5M :
1
.00
CD
F
(10-1
4 nb
GeV
6 )
Dat
a / C
TE
Q5M
CT
EQ
5HJ
/ CT
EQ
5MC
DF
Dat
a (
Pre
l. )
CT
EQ
5HJ
CT
EQ
5M
Co
mp
ariso
n o
f C
DF
incl
usi
ve je
t cr
oss
-se
ctio
n m
ea
sure
me
nt w
ithN
LO
QC
D c
alc
ula
tion
s w
ith C
TE
Q5
M a
nd
CT
EQ
5H
J P
DF
s
Inclusive Jet Production Data of D0
-- compared to NLO QCD
using CTEQ4HJ (CTEQ4M)
What's Uncertain about PDFs?
Quite a lot!
Strange and anti-strange Quarks
Details in the {u,d} quark sector: Up/Down differences and ratios (important for precision W/Z physics at colliders)
The Gluon Distribution (important for most SM and New Physics processes at very high energies)
Heavy Quark Distributions: Very little is known! Are heavy quarks “radiatively-generated"
exclusively; or are there intrinsic components of charm and/or bottom?
It is important to quantify the uncertainties of the parton distributions functions (PDF’s) and, more importantly, physical predictions which depend on PDF’s – not just the “bands”, but a systematic knowledge of the correlated variations.
������������ ���������������������������������������������� �����!��"���#$�&%&'(��)+* �����,����"��-/.$0/1�2�34�������������������65����������&78./"�8���������9��������:���,�������������������� �;�<���)=�"�,-/.$0/1�2>��� �?-/.$0/1;@9��� ����A�����(%���� �>�"�������"���,���B�<���)C�"�:39�;D�.E��� ����A������;FG��HB�HI��(JK7
L
Percentage Uncertainties of Parton
Distributions (By M. Botje)
Sources of uncertainties:
� Experimental statistical and systematic errors;
� \input": various theoretical corrections;
� \analysis": phenom. analysis procedure;
� \Scale": PQCD renor. and fact. scale-dependence
Figure 9: The parton momentum densities xg, xS (both divided by a factor of 20),xuv and xdv versus x at Q2 = 10 GeV2. The full curves show the results from the QCDfit with the errors drawn as shaded bands. The dashed curves are from the CTEQ4 [8]parton distribution set.
25
Global Analyses using an Effective χ2 Function
• Stress inclusion of all relevant (global) experi-mental data to constraint the PDFs;• Intend to be logical & useful extension of con-ventional, practical global QCD analyses.
⇒ Use an effective χ2-analysis with function χ2global
Use a collection of DIS experiments
S. Alekhin, hep-ph/0011002
V. Barone, C. Pascaud and F. Zomer,Eur. Phys. J. C12, 243 (2000) [hep-ph/9907512].
M. Botje,Eur. Phys. J. C14, 285 (2000) [hep-ph/9912439].
Use full global analysis data set of CTEQ5,+ some key advances in methodology
J. Pumplin, D. Stump,R. Brock, D. Casey, J. Huston, J. Kalk, WKT
(MSU)
⇒ hep-ph/0008191, 0101032, 0101052 ⇐
For physical variable X:Std.Min. set S0 : X
Alternate hypotheses sets {Si
±} i=1, ..2n: X
Bird's Eye View of the MSU Study(Pumplin, Stump, WKT, et al.)
Choice of priors:NLO QCD ;
other theory models;Parametrization of
nonperturbative PDFs.
Inputs:
Output:
Experimental Inputs:(i) full global data set of CTEQ5;(ii) effective global 2 function:
"Sampling"2 complementary optimized systematic methods:
ai
aj
2 - contours
Error Estimates2 distinct steps:(i) calc. CL in each expt.;(ii) estimate overall uncertainty.
(i) Lagrange multiplier;(ii) Hessian matrix.
2-dim illustration of the neighborhood of the globalminimum in the 16-dim parton parameter space
LX
Comparison of data and CTEQ5m (\theory")
-4 -2 0 2 4x=Hm-tL�Σ
0
200
400
600
800
dN�d
x
all data
The histogram includes all data used in the �t exceptjet production.
The curve has no adjustable parameters; it's just
N exp(�x2)=p2�
where N is the number of data points. The area underthe curve (or histogram) is N .
Di�erences mi�ti are within the published measurementerrors.
At least globally the distribution of uctuations is Gaus-
sian with the right width.
First Test of whether the effective global 2 method makesany sense: look at the distribution of the fluctuations of
the 1300 data points included in the global fit.
Lagrange Multiplier Method: (specific but robust)Let X be a physical quantity, then minimize Ψ =χ2g + λX to probe the neighborhood of the mini-mum.
ai
2-dim (i,j) renderingof d-dim PDF parameter space
contours of c2global
LM method
MC sampling
X: physicsvariable
LX
aj
• Sample variation ofχ2g as a function of X
by {SmX}, m = 1,2 . . .;
• Obtain PDF sets{S±
X} which extremizethe variation of X fora given tolerance ∆χ2
of χ2global.21 21.5 22 22.5
W nb
1200
1230
1260
1290
1320
1350
2gl
obal
W production at the Tevatron
W
T2
• Uncertainty of X: ∆X = 12(X(S
+X )− X(S−
X)),Apply to, e.g. σW/Z, σTop, σHiggs, . . . , at the Teva-
tron, RHIC and LHC.
Error Estimate: 1. Use all information provided by each individual experi-ments to calculate the (90%) CL w.r.t. that experiment.Example, H1 expt. (with full correlated error matrix):
21 21.5 22 22.5W nb Tevatron
0.98
1.02
1.06
1.1
H1
2N
This 2 functiondoes have statis-tical significance!The dashed hori-zontal line is the90% CL level.The error bar w.r.t.the H1 expt. is shownby the red line with arrows.
2. Make the 90% error bar calc. for all expts.
Tevatron
Wnb
20
21
22
23
24
25
BC
DM
Sp
BC
DM
Sd
H1
ZE
US
NM
Cp
NM
Cr
NM
Crx
CC
FR2
CC
FR3
E60
5
NA
51
CD
Fw
E86
6
D0j
et
CD
Fjet
W
In view of the problem with interpretation of the absolute 2 for some real experiments, use relative w.r.t. the best estimate solution S0.
Optimized Sampling of PDF Parameter Space– “Alternate Hypotheses” in uncertainty study
Hessian Matrix: (general but approx.)Diagonalize the Hessian matrix calculated from χ2g ,then move along each of the eigenvectors, i, to getup/down PDF sets {S±
i }.
(a)Original parameter basis
(b)Orthonormal eigenvector basis
zk
Tdiagonalization and
rescaling bythe iterative method
ul
ai
2-dim (i,j) rendition of d-dim (~16) PDF parameter space
Hessian eigenvector basis sets
ajul
p(i)
s0s0
contours of constant 2global
ul: eigenvector in the l-direction p(i): point of largest ai with tolerance T
s0: global minimum p(i)zl
· ⇒ PDF sets {S±i }, i = 1, . . . , n, spanning the
full PDF parameter space in the neighborhood ofthe global mimimum;
· The uncertainty of any physical variable X canbe calculated as: (the master eq.)
∆X =T
2t
√∑i
(X(S+i )− X(S−i ))
2
7KH�+HVVLDQ�0HWKRG�RI�TXDQWLI\LQJ�XQFHUWDLQWLHV�E\�D�FRPSOHWH�VHW�RI�RUWKRQRUPDO HLJHQYHFWRU�3')V
Global landscape:- physics* gradients vary widely:steep / flatness measured by eigen values of Hessian: ~ 106
Local bumpiness: unphysical various reasons: theory model is NOT smooth!;finite steps, MC methods, ...
Challenges to Numerical Calculation of the Hessian Matrix
in 16-dimensional PDF parameter space
ai
aj
It is well known that general purpose minimization programs do not provide reliable error estimates for global analysis. (Step size choices not designed to deal with these problems)
Eigenvalues of the Hessian Matrixfor two different choices of parametrization
of the PDF parameter space
Iterative Method to generate Eigenvectors:(and dramatically improve numerical reliability)
ai0
aj0
Physical parameters
aiN = zi
ajN = zj
Nth iteration:Eigenvectororthonormalbasis
ain
ajn
nth iteration
the χ2 = const. ellipsoid
jijij
i aXH
aXX
∂∂
�∂∂∆=∆ − )( 122 χ
2222 )]()([)( −+� −=� ∂
∂∆=∆ ii iii
SXSXzXX χ
Si+Si
−
Sj+
Sj−
* This iterative method has been shown by Botje and Zomer to greatly improve their own global analyses, and it is now being considered for adoption in MINUIT (as an option) by its author F. James.
Considerations in Estimating the Tolerance T of χglobal2
(∆χglobal2 < T2)
• ∆χglobal2 = 1 has no statistical significance in this
context. Most experiments only give a combined (“effective”) systematic error; Combining χ2 ’s of 15 experiments on very diverse processes and accuracies is an extremely dicey business.
• Basic assumption of 15 acceptable experiments, in spite of the fact that some are nominally inconsistent in the strict statistical sense, implies T >> 1.
• Basic assumption of compatibility of the 15 experiments, in spite of the fact that some are nominally incompatible in the strict statistical sense, implies T >> 1. We can “measure” the acceptable T for compatibility.
• Quantitative estimate of T from comparison of sample PDFs (the Alternate Hypotheses) obtained from Hessian and Lagrange methods to the individual experiments. o Evaluate the 90% CL range for each experiment based
on available error information (using χ2 normalized to standard fit, if necessary)
o Combine these ranges into one estimated tolerance for Τ Bottom Line:
10 < T < 15
Two
ext
rem
e p
art
on
dis
trib
utio
ns
with
T =
10
at Q
= 2
Ge
V a
nd
Q
= 1
00
G
eV
(in
da
she
d li
ne
s). T
he
un
cert
ain
ty b
an
d fo
rT
< 1
0 is
sh
ad
ed
.(T
de
fine
s th
e tole
ranc
e:
2 glo
ba
l < T
2)
Glu
onU
p qu
ark
Ra
tio o
f p
art
on
dis
trib
utio
n to
Be
st F
it d
istr
ibu
tion
for
two
ext
rem
e T
= 1
0 c
ase
s a
t Q
= 1
0 G
eV.
Th
e r
egio
n a
llow
ed
by
T <
10
is s
ha
de
d.
Glu
on
Up
-qu
ark
Predicted rapidity distribution for W production at the Tevatron
Six curves represent the alternate hypotheses of one± 2 for the physical variables t, <y>, and <y2>.
Predicted rapidity distribution for W production at the Tevatron-- deviation from the best estimate solution S0
Six curves represent the alternate hypotheses of one± 2 for the physical variables t, <y>, and <y2>.
Predicted rapidity distribution for W production at the Tevatron-- deviation from the best estimate solution S0
Three (of the six) curves for each case representing the"extremes" for the physical variables t, <y>, and <y2>.
Compare the results obtained by theLagrange and Hessian methods
Since the LM method is fully robust (no approx.), the agreement proves the efficacy of the the Hessian method.
Correlated uncertainties in W/Z productionCross-section at the Tevatron
CDF
D0
Ellipse : Allowed region with Tolerance = 10.
** CDF with the same luminosity input as D0
**
102 103
Sqrt (s^)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fra
ctio
nal U
ncer
tain
ties
at Tevatron Run IIFractional Luminosity Uncertainties
W, Z
GGQQbar γQQbar W+
QQbar ZGQ γGQ W-
102
Sqrt (s^)
0
0.1
0.2
0.3
Fra
ctio
nal U
ncer
tain
ties
at Tevatron Run IIFractional Luminosity Uncertainties
W, Z 115 GeV
GGQQbar γQQbar W+
QQbar ZGQ γGQ W-
102 103
sqrt (s^)
0
0.05
0.1
0.15
0.2
0.25
Fra
ctio
nal U
ncer
tain
ties
W, Z 115 GeV
Luminosity Uncertainties at LHC
GGQQbar γQQbar W+
GQ W+
GQ Z
101 102
sqrt (s^)
0
0.1
0.2
0.3
0.4
0.5
0.6
Fra
ctio
nal U
ncer
tain
ties
W, Z
Luminosity Uncertainties at RHIC 500 GeV
GGQQbar γQQbar W+
GQ W+
GQ W-
101 102
sqrt (s^)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fra
ctio
nal U
ncer
tain
ties
W, Z
Luminosity Uncertainties at RHIC 200 GeV
GGQQbar γQQbar W+
QQbar W-
GQ γGQ W+
Outlook
This is only the very beginning of studying uncertainties in global QCD analysis in a quantitative manner It should be regarded more as a demonstration of
principles. There is a lot of room for collaboration among
theorists and experimentalists
Several data sets are about to be updated (CCFR, H1, ZEUS, ... ) � Specific results of this study will likely change soon.
Many other sources of uncertainties in the overall global analysis have not yet been incorporated: Theoretical uncertainties due to higher-order PQCD
corrections and resummation; Uncertainties introduced by the choice of parametrization
(This has been partially explored by us.) The Hessian eigenvectors provide a systematic way to distinguish “flat/steep” directions in parameter space; and to suggest ways to improve the choice of parametrization.
Continued progress in this venture is of vital importance for our understanding of the parton structure of hadrons (fundamental physics of its own right), for precision SM physics studies at future colliders, and for New Physics searches.